Consistently improving approximations for constrained controllability and reachability

This paper develops a method for the construction of consistently improving inner and outer approximate poly-topic sets for controllability and reachability under constraints. Sequences of consistently improving approximate polytopic sets are obtained from the solution of a series of Second Order Cone Programming (SOCP) problems. Notable advances in reliability and speed of recent SOCP solvers render the proposed procedure computationally attractive, since the solutions to frequently encountered SOCPs are computable in tens of milliseconds or less. The initialization of the previous algorithm is enhanced by means of a simultaneous computation of an optimized pair of inner and outer bounding simplices. The Hausdorff distance is employed as a criteria for the growth of the inner and outer approximate polytopic sets as well as a topologically suitable metric for convergence. The algorithm yields polytopes that inner and outer approximate exact Controllable/Reachable (C/R) set to a desired precision w.r.t. the Hausdorff distance. The relevance and value of the proposed procedure are demonstrated by its application to a fully constrained Mars planetary landing problem.